Find value of y

Question

log(-1+yi)=ln 5.10+1.77i

Answer 1

Hello Wendy, I hope you're doing well. Allow me to make 2 notes first: 1) I checked both answers from my fellow tutors and have verified that neither answer is correct due to an algebraic mistake: they assumed that ln(a)/ln(b) = ln(a/b) which is of course incorrect. (Somewhat of a shocking mistake if you ask me, and honestly makes me question their credentials to be a math tutor.) 2) After attempting the following problem, I can see that the value of 1.77 was rounded to the nearest hundredth.

We start with log(-1+yi) = ln(5.1) +1.77i. We must first understand, that both logarithms have the same base of e.


First, by definition of a complex log: log (a+bi) = ln( sqrt (a^2 + b^2)) + i*arctan(b/a).

Therefore log(-1+yi) = ln( sqrt ( (-1)^2 + y^2)) + i*arctan(-y) which equals ln(5.1) + 1.77i by the given equation.

Therefore sqrt ( (-1)^2+y^2) = 5.1. After solving for y, we obtain exactly 5.0009999. If we plug this value of y into

log(-1+yi) (remember that this is base e), we obtain approximately ln(5.1) + 1.768i which is rounded to ln(5.1) + 1.77i.

Hope this helps. All the best!

Answer 2

It's not clear what is inside of the ln on the right hand side. I assume the problem is:

 

log(-1+yi) = ln(5.10+1.77i)

 

Change of log base (log(a) = ln(a)/ln(10)):

 

ln(-1+yi)/ln(10) = ln(5.10+1.77i)

ln((-1+yi)/10) = ln(5.10+1.77i)

 

(-1 + yi)/10 = 5.1 + 1.77i

-1 + yi = 51 + 17.7i

yi = 52 + 17.7i

y = (52 + 17.7i)/i

 

Multiply the right side by i/i:

 

y = (52 + 17.7i/i) · (i/i)

y = (52i + 17.7i²)/i²

y = (52i - 17.7)/(-1)

 

y = 17.7 - 52i